In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial.
For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal.
This article uses the language of group theory; analogous terms are used for Lie algebras.
A related but distinct construction is the derived series, which terminates in the trivial subgroup whenever the group is solvable.
denotes the commutator subgroup generated by all elements of the form
A central series is analogous in Lie theory to a flag that is strictly preserved by the adjoint action (more prosaically, a basis in which each element is represented by a strictly upper triangular matrix); compare Engel's theorem.
Continuing in this way to choose the largest possible Ai + 1 given Ai produces what is called the upper central series.
A nilpotent group is a solvable group, and its derived length is logarithmic in its nilpotency class (Schenkman 1975, p. 201,216).
For infinite groups, one can continue the lower central series to infinite ordinal numbers via transfinite recursion: for a limit ordinal λ, define If
is the smallest normal subgroup of G such that the quotient is residually nilpotent, that is, such that every non-identity element has a non-identity homomorphic image in a nilpotent group (Schenkman 1975, p. 175,183).
In the field of combinatorial group theory, it is an important and early result that free groups are residually nilpotent.
In fact the quotients of the lower central series are free abelian groups with a natural basis defined by basic commutators, (Hall 1959, Ch.
is the smallest normal subgroup of G with nilpotent quotient, and
is called the nilpotent residual of G. This is always the case for a finite group, and defines the
There is no general term for the intersection of all terms of the transfinite lower central series, analogous to the hypercenter (below).
The upper central series (or ascending central series) of a group G is the sequence of subgroups where each successive group is defined by: and is called the ith center of G (respectively, second center, third center, etc.).
, and is called an upper central series quotient.
Again, we say the series terminates if it stabilizes into a chain of equalities, and its length is the number of distinct groups in it.
For infinite groups, one can continue the upper central series to infinite ordinal numbers via transfinite recursion: for a limit ordinal λ, define The limit of this process (the union of the higher centers) is called the hypercenter of the group.
Hypercentral groups enjoy many properties of nilpotent groups, such as the normalizer condition (the normalizer of a proper subgroup properly contains the subgroup), elements of coprime order commute, and periodic hypercentral groups are the direct sum of their Sylow p-subgroups (Schenkman 1975, Ch.
There are various connections between the lower central series (LCS) and upper central series (UCS) (Ellis 2001), particularly for nilpotent groups.
For a nilpotent group, the lengths of the LCS and the UCS agree, and this length is called the nilpotency class of the group.
However, the LCS and UCS of a nilpotent group may not necessarily have the same terms.
A group is abelian if and only if the LCS terminates at the first step (the commutator subgroup is the trivial subgroup), if and only if the UCS terminates at the first step (the center is the entire group).
By contrast, the LCS terminates at the zeroth step if and only if the group is perfect (the commutator is the entire group), while the UCS terminates at the zeroth step if and only if the group is centerless (trivial center), which are distinct concepts.
For a perfect group, the UCS always stabilizes by the first step (Grün's lemma).
However, a centerless group may have a very long LCS: a free group on two or more generators is centerless, but its LCS does not stabilize until the first infinite ordinal.
This shows that the lengths of the LCS and UCS need not agree in general.
In the study of p-groups (which are always nilpotent), it is often important to use longer central series.
There is a unique most quickly ascending such series, the upper exponent-p central series S defined by: where Ω(Z(H)) denotes the subgroup generated by (and equal to) the set of central elements of H of order dividing p. The first term, S1(G), is the subgroup generated by the minimal normal subgroups and so is equal to the socle of G. For this reason the upper exponent-p central series is sometimes known as the socle series or even the Loewy series, though the latter is usually used to indicate a descending series.